Sample complexity of sample average approximation for conditional stochastic optimization

In this paper, we study a class of stochastic optimization problems, referred to as the conditional stochastic optimization (CSO), in the form of minx∈X Eξ fξ (Eη ξ [gη (x, ξ)]), which finds a wide spectrum of applications including portfolio selection, reinforcement learning, robust learning, and causal inference. Assuming availability of samples from the distribution P(ξ) and samples from the conditional distribution P(η ξ), we establish the sample complexity of the sample average approximation (SAA) for CSO, under a variety of structural assumptions, such as Lipschitz continuity, smoothness, and error bound conditions. We show that the total sample complexity improves from O (d/ 4) to O (d/ 3) when assuming smoothness of the outer function, and further to O (1/ 2) when the empirical function satisfies the quadratic growth condition. We also establish the sample complexity of a modified SAA when ξ and η are independent. Several numerical experiments further support our theoretical findings.